(* 4.382.3 *)

Integrate[Log[(a^2+x^2)/(b^2+x^2)]*Cos[c*x],{x,0,Infinity},Assumptions->(a>0)&&(b>0)&&(c>0)]//FullSimplify

(* Mathematica can not do this. Now separate the logarithms *)

Integrate[Log[(a^2+x^2)]*Cos[c*x]-Log[(b^2+x^2)]*Cos[c*x],{x,0,p},Assumptions->(a>0)&&(b>0)&&(c>0)&&(p>0)]//FullSimplify

(* This could not be simplified *)